Question: Simplify and expand the following expression: $ \dfrac{5p + 6}{5p + 1}-\dfrac{p}{p - 5} $
Explanation: In order to subtract expressions, they must have a common denominator. Get both fractions over a common denominator of $(5p + 1)(p - 5)$ Multiply the first term by $\dfrac{p - 5}{p - 5}$ $ \begin{align*} \dfrac{5p + 6}{5p + 1} \times \dfrac{p - 5}{p - 5} & = \dfrac{(5p + 6)(p - 5)}{(5p + 1)(p - 5)} \\ & = \dfrac{5p^2 - 19p - 30}{(5p + 1)(p - 5)}\end{align*} $ Multiply the second term by $\dfrac{5p + 1}{5p + 1}$ $ \begin{align*} \dfrac{p}{p - 5} \times \dfrac{5p + 1}{5p + 1} & = \dfrac{(p)(5p + 1)}{(p - 5)(5p + 1)} \\ & = \dfrac{5p^2 + p}{(p - 5)(5p + 1)}\end{align*} $ Now we have: $ = \dfrac{5p^2 - 19p - 30}{(5p + 1)(p - 5)} - \dfrac{5p^2 + p}{(p - 5)(5p + 1)} $ Now both terms have a common denominator we can subtract the numerators: $ = \dfrac{5p^2 - 19p - 30 - (5p^2 + p)}{(5p + 1)(p - 5)} $ $ = \dfrac{5p^2 - 19p - 30 - 5p^2 - p}{(5p + 1)(p - 5)} $ $ = \dfrac{-20p - 30}{(5p + 1)(p - 5)}$ Expand the denominator: $ = \dfrac{-20p - 30}{5p^2 - 24p - 5}$